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Author : E. B. Davies
Publisher : Cambridge University Press
Page : 212 pages
File Size : 54,5 Mb
Release : 1989
Category : Mathematics
ISBN : 0521409977
Heat Kernels and Spectral Theory investigates the theory of second-order elliptic operators.
Author : E. Brian Davies
Publisher : Cambridge University Press
Page : 198 pages
File Size : 45,7 Mb
Release : 1995
Category : Mathematics
ISBN : 0521587107
This book could be used either for self-study or as a course text, and aims to lead the reader to the more advanced literature on partial differential operators.
Author : Alexander Grigor'yan
Publisher : American Mathematical Soc.
Page : 504 pages
File Size : 46,8 Mb
Release : 2009
Category : Gaussian processes
ISBN : 9780821893937
The heat kernel has long been an essential tool in both classical and modern mathematics but has become especially important in geometric analysis as a result of major innovations beginning in the 1970s. The methods based on heat kernels have been used in areas as diverse as analysis, geometry, and probability, as well as in physics. This book is a comprehensive introduction to heat kernel techniques in the setting of Riemannian manifolds, which inevitably involves analysis of the Laplace-Beltrami operator and the associated heat equation. The first ten chapters cover the foundations of the subject, while later chapters deal with more advanced results involving the heat kernel in a variety of settings. The exposition starts with an elementary introduction to Riemannian geometry, proceeds with a thorough study of the spectral-theoretic, Markovian, and smoothness properties of the Laplace and heat equations on Riemannian manifolds, and concludes with Gaussian estimates of heat kernels. Grigor'yan has written this book with the student in mind, in particular by including over 400 exercises. The text will serve as a bridge between basic results and current research.Titles in this series are co-published with International Press, Cambridge, MA, USA.
Author : Michael Demuth,Werner Kirsch
Publisher : Springer Science & Business Media
Page : 339 pages
File Size : 48,7 Mb
Release : 2014-07-08
Category : Mathematics
ISBN : 9783034805919
This volume presents self-contained survey articles on modern research areas written by experts in their fields. The topics are located at the interface of spectral theory, theory of partial differential operators, stochastic analysis, and mathematical physics. The articles are accessible to graduate students and researches from other fields of mathematics or physics while also being of value to experts, as they report on the state of the art in the respective fields.
Author : Fan R. K. Chung
Publisher : American Mathematical Soc.
Page : 228 pages
File Size : 42,6 Mb
Release : 1997
Category : Eigenvalues
ISBN : 9780821803158
This text discusses spectral graph theory.
Author : M. Demuth,B.W. Schulze,B. Gramsch
Publisher : Birkhäuser
Page : 355 pages
File Size : 55,9 Mb
Release : 2012-12-06
Category : Science
ISBN : 9783034886239
Author : Fengyu Wang
Publisher : Elsevier
Page : 391 pages
File Size : 40,8 Mb
Release : 2006-04-06
Category : Mathematics
ISBN : 9780080532073
In this book, the functional inequalities are introduced to describe:(i) the spectrum of the generator: the essential and discrete spectrums, high order eigenvalues, the principle eigenvalue, and the spectral gap;(ii) the semigroup properties: the uniform intergrability, the compactness, the convergence rate, and the existence of density;(iii) the reference measure and the intrinsic metric: the concentration, the isoperimetic inequality, and the transportation cost inequality.
Author : Jay Jorgenson,American Mathematical Society
Publisher : American Mathematical Soc.
Page : 410 pages
File Size : 40,7 Mb
Release : 2006
Category : Geometry, Algebraic
ISBN : 9780821836989
The aim of this volume is to bring together research ideas from various fields of mathematics which utilize the heat kernel or heat kernel techniques in their research. The intention of this collection of papers is to broaden productive communication across mathematical sub-disciplines and to provide a vehicle which would allow experts in one field to initiate research with individuals in another field, as well as to give non-experts a resource which can facilitate expanding theirresearch and connecting with others.
Author : Nicole Berline,Ezra Getzler,Michèle Vergne
Publisher : Springer Science & Business Media
Page : 384 pages
File Size : 40,5 Mb
Release : 2003-12-08
Category : Mathematics
ISBN : 3540200622
In the first edition of this book, simple proofs of the Atiyah-Singer Index Theorem for Dirac operators on compact Riemannian manifolds and its generalizations (due to the authors and J.-M. Bismut) were presented, using an explicit geometric construction of the heat kernel of a generalized Dirac operator; the new edition makes this popular book available to students and researchers in an attractive paperback.
Author : Jun Kigami
Publisher : American Mathematical Soc.
Page : 110 pages
File Size : 44,5 Mb
Release : 2009-04-10
Category : Mathematics
ISBN : 9780821842928
This paper studies the following three problems. 1. When does a measure on a self-similar set have the volume doubling property with respect to a given distance? 2. Is there any distance on a self-similar set under which the contraction mappings have the prescribed values of contractions ratios? 3. When does a heat kernel on a self-similar set associated with a self-similar Dirichlet form satisfy the Li-Yau type sub-Gaussian diagonal estimate? These three problems turn out to be closely related. The author introduces a new class of self-similar set, called rationally ramified self-similar sets containing both the Sierpinski gasket and the (higher dimensional) Sierpinski carpet and gives complete solutions of the above three problems for this class. In particular, the volume doubling property is shown to be equivalent to the upper Li-Yau type sub-Gaussian diagonal estimate of a heat kernel.
Author : David Borthwick
Publisher : Springer Nature
Page : 339 pages
File Size : 41,8 Mb
Release : 2020-03-12
Category : Mathematics
ISBN : 9783030380021
This textbook offers a concise introduction to spectral theory, designed for newcomers to functional analysis. Curating the content carefully, the author builds to a proof of the spectral theorem in the early part of the book. Subsequent chapters illustrate a variety of application areas, exploring key examples in detail. Readers looking to delve further into specialized topics will find ample references to classic and recent literature. Beginning with a brief introduction to functional analysis, the text focuses on unbounded operators and separable Hilbert spaces as the essential tools needed for the subsequent theory. A thorough discussion of the concepts of spectrum and resolvent follows, leading to a complete proof of the spectral theorem for unbounded self-adjoint operators. Applications of spectral theory to differential operators comprise the remaining four chapters. These chapters introduce the Dirichlet Laplacian operator, Schrödinger operators, operators on graphs, and the spectral theory of Riemannian manifolds. Spectral Theory offers a uniquely accessible introduction to ideas that invite further study in any number of different directions. A background in real and complex analysis is assumed; the author presents the requisite tools from functional analysis within the text. This introductory treatment would suit a functional analysis course intended as a pathway to linear PDE theory. Independent later chapters allow for flexibility in selecting applications to suit specific interests within a one-semester course.
Author : Pascal Auscher,T. Coulhon,Alexander Grigoryan
Publisher : American Mathematical Soc.
Page : 434 pages
File Size : 40,6 Mb
Release : 2003
Category : Elliptic operators
ISBN : 9780821833834
This volume contains the expanded lecture notes of courses taught at the Emile Borel Centre of the Henri Poincare Institute (Paris). In the book, leading experts introduce recent research in their fields. The unifying theme is the study of heat kernels in various situations using related geometric and analytic tools. Topics include analysis of complex-coefficient elliptic operators, diffusions on fractals and on infinite-dimensional groups, heat kernel and isoperimetry on Riemannian manifolds, heat kernels and infinite dimensional analysis, diffusions and Sobolev-type spaces on metric spaces, quasi-regular mappings and $p$-Laplace operators, heat kernel and spherical inversion on $SL 2(C)$, random walks and spectral geometry on crystal lattices, isoperimetric and isocapacitary inequalities, and generating function techniques for random walks on graphs. This volume is suitable for graduate students and research mathematicians interested in random processes and analysis on manifolds.
Author : Ivan Avramidi
Publisher : Birkhäuser
Page : 390 pages
File Size : 51,9 Mb
Release : 2015-11-26
Category : Mathematics
ISBN : 9783319262666
The heart of the book is the development of a short-time asymptotic expansion for the heat kernel. This is explained in detail and explicit examples of some advanced calculations are given. In addition some advanced methods and extensions, including path integrals, jump diffusion and others are presented. The book consists of four parts: Analysis, Geometry, Perturbations and Applications. The first part shortly reviews of some background material and gives an introduction to PDEs. The second part is devoted to a short introduction to various aspects of differential geometry that will be needed later. The third part and heart of the book presents a systematic development of effective methods for various approximation schemes for parabolic differential equations. The last part is devoted to applications in financial mathematics, in particular, stochastic differential equations. Although this book is intended for advanced undergraduate or beginning graduate students in, it should also provide a useful reference for professional physicists, applied mathematicians as well as quantitative analysts with an interest in PDEs.
Author : Jun Kigami
Publisher : American Mathematical Soc.
Page : 145 pages
File Size : 51,8 Mb
Release : 2012-02-22
Category : Mathematics
ISBN : 9780821852996
Assume that there is some analytic structure, a differential equation or a stochastic process for example, on a metric space. To describe asymptotic behaviors of analytic objects, the original metric of the space may not be the best one. Every now and then one can construct a better metric which is somehow ``intrinsic'' with respect to the analytic structure and under which asymptotic behaviors of the analytic objects have nice expressions. The problem is when and how one can find such a metric. In this paper, the author considers the above problem in the case of stochastic processes associated with Dirichlet forms derived from resistance forms. The author's main concerns are the following two problems: (I) When and how to find a metric which is suitable for describing asymptotic behaviors of the heat kernels associated with such processes. (II) What kind of requirement for jumps of a process is necessary to ensure good asymptotic behaviors of the heat kernels associated with such processes.
Author : Wolfgang Arendt,Joseph A. Ball,Jussi Behrndt,Karl-Heinz Förster,Volker Mehrmann,Carsten Trunk
Publisher : Springer Science & Business Media
Page : 684 pages
File Size : 48,5 Mb
Release : 2012-06-15
Category : Mathematics
ISBN : 9783034802970
The present volume contains a collection of original research articles and expository contributions on recent developments in operator theory and its multifaceted applications. They cover a wide range of themes from the IWOTA 2010 conference held at the TU Berlin, Germany, including spectral theory, function spaces, mathematical system theory, evolution equations and semigroups, and differential and difference operators. The book encompasses new trends and various modern topics in operator theory, and serves as a useful source of information to mathematicians, scientists and engineers.
